Band Theory of Solids
Band Theory of Solids
Band Theory of Solids
. . . . .
a a
system
2S
2p
a r
The allowed energy levels for an atom are discrete (2 electrons
with opposite spin can occupy a state)
When atoms are brought into close contact, these energy levels
split
If there are a large number of atoms, the discrete energy levels
form a “continuous” band.
Hence, at equilibrium interatomic spacing, energy bands are
formed in a solid.
Due to the discrete nature of energy , there are forbidden gaps in between the
Many atomic system—a solid
V(X)
3p
3s
2p
-Ze2/ 4πε0r
2s
1s
a
X
Periodic Potential: V (x) = V( x +n a )
E3
E2
E
E1
r
E3
E E2
E1
E
r0
r0 r
This also leads to lowering of total energy.
but the energy levels of inner electrons
remain sharp
The width of the bands depend on E
interatomic spacing
E3
E3
E
E2 E2
E1 E1
r0 r
Higher bands are wider and forbidden gap decreases towards
higher state
When no. of free electrons are more, splitting is more, so,
overlapping of bands occur
Depending on the no. of electrons in a solid, some bands are
completely filled, some partially filled and some are empty
Highest filled band is called valence band and lowest empty band
is called conduction band.
Most of the properties can be understood by considering the
VB,CB and the FG between them.
These are… the occupation state of VB, width of FG, density of
state in the VB and the impurities present which introduces
allowed states in the FG
Conduction band
Forbidden gap
Valence band
•Valence band may be
partially filled, half filled or
completely filled
Eg
EC
EV
EC
Eg = 3 - 7 eV
Mg12 Eg = 1eV
EC EV
EV
EV
Conductors: These solids have either half filled valence band like
Cu, so that, no. of free electrons available is exactly equal to the
no. of allowed states available in the same band, a very small
amount of externally supplied energy can enable the electrons to
move raise the conductivity
Or, some have overlapping of valence band and conduction band,
such that there is no distinguished barriers for the electrons to
overcome. On the other hand, plenty of allowed states are
available for the electrons, due to the merging of two bands.
Ex:- Mg
Insulators:
• Valence band completely filled, conduction band empty and
forbidden gap very wide (3—7 eV),
• Normally electrons can not acquire this large energy to overcome
the forbidden gap and remain immobile.
Semiconductors:
• Filled valence band
• Empty conduction band
• Narrow forbidden gap (1 eV)
• Electrons in the valence band can easily acquire this energy, ( at
T>0 or ext. electric field )free themselves from the valence band
and move in the conduction band, behaving as free carriers.
• When the electrons leave the valence band, a vacant space—
”hole” is created in the valence band which are equivalent to
positive charge carriers and contribute to the conductivity.
conductivity :=
. . .. . Free electrons
σ = neμ e + pe μ h
o o o o o Free holes
Mathematical approach– Kronig-Penny model :
• Electrons moving inside the solid material experience a periodic
potential
• V (x) = V (x+ na)
• The Schrödinger's equation for the electrons is
2
( x) 2m
• (E V) ( x) 0
x2 2
• Where, uk ( x) uk ( x na)
• Is called the Bloch’s function having the same periodicity “a”
• Solution for Schrödinger's equation for Kronig Penny model is
possible for energies that satisfy the following conditions
V(x)
Barrier
(II)
Well
(I)
- (a+b) -b 0 a a+b
P
sin a cos a cos ka
a
Where, P= β2ab / 2 , β2= 2m(V0-E) / ħ2 , α2= 2mE / ħ2
+1
a
-1
α 2= 2mE / ħ 2
band gap
1D
2nd Brillouin
zone
2nd Brillouin
zone
1st Brillouin
zone
-2π / a
2D
Intrinsic and extrinsic semiconductor
• conductivity of pure semiconductors is poor
• doping improves the conductivity
pentavalent impurity added to pure SC results in n-type SC ,
where, majority carriers are electrons and minority are holes,
conductivity is σ =neμe
whereas trivalent impurity addition results in the p-type SC ,
majority carrier holes and minority are electrons
conductivity σ =neμh
Fermi level
pure n-type p-type
Ef
Ef
Ef
When a current carrying conductor/
Hall Effect: semi-Conductor is placed in an ext.
IX magnetic Field a potential difference is
established in a direction perpendicular
to both Current and magnetic field,
known as Hall voltage
current Ix in the material along x-axis
magnetic field Bz along z-axis
Bz current density Jx = nq <Vx>
Lorentz force on electrons due to
N S magnetic field,
F= Fy= q ( vXB) or Fy= q<vx>Bz
VHY Due to charge separation, a potential
difference (electric field ) is set up
Or, qEH= q<vx>Bz
Or, EH = <vx>Bz= Jx Bz/ nq
As, VH= Ehw
Hence, (VH)y= IxBzw / nq wt
Or, Hall voltage, (VH)y = RH IxBz / t
Since, in a semiconductor, two types of
Where RH= 1 / nq,, is known as Hall co-
Charge carriers are present, RH can be
efficient
positive(p-type) or negative (n-type)